R84 练习: Portfolio Risk and Return Part II

R84 练习: 组合风险与收益（下）

考纲范围

• Describe the implications of combining a risk-free asset with a portfolio of risky assets. 描述：无风险资产与风险资产构成的投资组合的含义
• Explain the capital allocation line (CAL) and the capital market line (CML). 说明：资本配置线（CAL）和资本市场线（CML）
• Explain systematic and nonsystematic risk, including why an investor should not expect to receive additional return for bearing nonsystematic risk. 解释：系统性和非系统性风险（包括为什么投资者不会因为承担非系统风险而获得额外回报）
• Explain return generating models (including the market model) and their uses. 说明：产生回报的模型（包括市场模型）及其用途
• Calculate and interpret beta. 计算/解释：beta
• Explain the capital asset pricing model (CAPM), including its assumptions, and the security market line (SML). 解释：资本资产定价模型（CAPM），包括其假设和证券市场线（SML）
• Calculate and interpret the expected return of an asset using the CAPM. 计算/解释：资产的预期回报（使用 CAPM）
• Describe and demonstrate applications of the CAPM and the SML. 描述/演示：CAPM 和 SML 的应用
• Calculate and interpret the Sharpe ratio, Treynor ratio, M2, and Jensen’s alpha. 计算/解释：夏普比率，特雷诺比率，M2 和詹森 alpha

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Q1.

A portfolio is constructed with 40% risk-free asset and 60% risky asset. If the standard deviation of the risky asset return is 10%, the variance of the portfolio is closest to:

A. 0.0036.

B. 0.0016.

C. 0.0025.

查看答案与解析

答案：A

解析：无风险资产的标准差为零，与风险资产的协方差也为零。

计算过程：

${\sigma }_p=w_{risky}\times {\sigma }_{risky}=0.60\times 0.10=0.06$

${\sigma }_p^2=(0.06{)}^2=0.0036$

选项	判断	解析
A	✓	$(0.60\times 0.10{)}^2=0.0036$
B	✗	可能错误地使用了 40% 的权重
C	✗	可能错误地使用了 50% 的权重

关联：R84: Portfolio Risk and Return II

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Q2.

The plot of capital allocation line is a straight line which intersects with y axis at a point. This point most likely refers to:

A. risky asset.

B. risk-free asset.

C. preferable portfolio.

查看答案与解析

答案：B

解析：资本配置线（CAL）是从无风险利率出发，通过最优风险组合的直线。CAL与Y轴（预期回报轴）的交点代表100%投资于无风险资产的情况，即无风险利率点。此时标准差为零，预期回报等于无风险利率。

选项	判断	解析
A	✗	风险资产位于CAL的某个位置，但不在Y轴交点
B	✓	Y轴交点代表100%无风险资产投资，标准差为零
C	✗	偏好组合取决于投资者，不一定在Y轴交点

关联：R84: Portfolio Risk and Return II

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Q3.

Investor A is more risk averse than Investor B. Compared with Investor B, Investor A’s optimal portfolio has:

A. a higher weight in the optimal risky portfolio.

B. a higher weight in the risk-free asset.

C. the same weights in both risk-free asset and the optimal risky portfolio.

查看答案与解析

答案：B

解析：根据两基金分离定理（Two-fund separation theorem），所有投资者持有相同的最优风险组合，只是在无风险资产和最优风险组合之间的配置比例不同。风险厌恶程度较高的投资者A会将更多资金配置于无风险资产，减少风险资产的权重。

选项	判断	解析
A	✗	更厌恶风险的投资者A在风险组合中的权重更低
B	✓	投资者A更厌恶风险，因此在无风险资产中的权重更高
C	✗	不同风险偏好的投资者配置比例不同

关联：R84: Portfolio Risk and Return II

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Q4.

There are two comments about the capital market line (CML).

Comment 1: The capital market line (CML) is a special form of capital allocation line (CAL).

Comment 2: CML is determined by the market portfolio and the coordinate origin which represents the risk-free asset.

Are the two comments correct?

A. Yes

B. No, comment 1 is wrong.

C. No, comment 2 is wrong.

查看答案与解析

答案：C

解析：Comment 1 正确：CML是CAL的特殊形式，当所有投资者对预期收益、方差和协方差有相同预期时，最优风险组合就是市场组合，此时CAL即为CML。Comment 2 错误：CML的起点不是坐标原点，而是Y轴上的无风险利率点（Rf），CML由无风险利率和市场组合确定。

选项	判断	解析
A	✗	Comment 2 是错误的
B	✗	Comment 1 是正确的
C	✓	Comment 2 错误：CML起点是Rf（不是坐标原点），通过市场组合

关联：R84: Portfolio Risk and Return II

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Q5.

With respect to capital market theory, an analyst conducted the following statements:

Statement 1: Capital market line (CML) is a special security market line (SML).

Statement 2: The theory assumes that all investors have a homogeneous expectation.

Statement 3: The market portfolio contains all assets.

The most accurate statement is:

A. statement 1.

B. statement 2.

C. statement 3.

查看答案与解析

答案：B

解析：Statement 1 错误：CML不是特殊的SML；两者是不同的概念（CML衡量总风险，SML衡量系统风险）。Statement 2 正确：CAPM的核心假设之一是所有投资者具有同质预期。Statement 3 需要注意：市场组合包含所有可投资（risky）资产，但这里说”all assets”可能含义模糊，严格来说市场组合包含所有风险资产。

选项	判断	解析
A	✗	CML 不是 SML 的特殊形式；两者维度不同（总风险 vs 系统风险）
B	✓	同质预期是资本市场理论的核心假设
C	✗	市场组合包含所有风险资产，但”all assets”表述不够精确

关联：R84: Portfolio Risk and Return II

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Q6.

An investor chooses a portfolio which lies on the Capital Market Line (CML) and at the right side of the market portfolio. The investor

A. achieves more diversification benefits.

B. finds it is more efficient than the market portfolio.

C. uses some leverage to construct the portfolio.

查看答案与解析

答案：C

解析：在CML上，市场组合右侧的点代表借入组合（borrowing portfolio）。投资者以无风险利率借入资金，将自有资金和借入资金全部投入市场组合，使用了杠杆。该组合具有更高的预期回报和更高的风险。

选项	判断	解析
A	✗	CML上的所有点都是充分分散化的有效组合，右侧并不比市场组合更分散
B	✗	CML上所有点都是有效的，不存在比市场组合”更有效”的说法
C	✓	市场组合右侧意味着借入资金投入市场组合，即使用了杠杆

关联：R84: Portfolio Risk and Return II

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Q7.

Which of the following statements about systematic risk and nonsystematic risk is correct?

Statement 1: Systematic risk can be diversified by constructing a diversified portfolio.

Statement 2: Investors are compensated for bearing nonsystematic risk.

A. Statement 1.

B. Statement 2.

C. None of them.

查看答案与解析

答案：C

解析：系统性风险（市场风险）不能通过分散化消除，只有非系统性风险可以通过分散化消除。因为非系统性风险可以被分散掉，市场不会为承担非系统性风险提供额外补偿。两个陈述都是错误的。

选项	判断	解析
A	✗	Statement 1 错误：系统性风险不能通过分散化消除（非系统性风险才可以）
B	✗	Statement 2 错误：市场只补偿系统性风险，不补偿非系统性风险
C	✓	两个陈述都是错误的

关联：R84: Portfolio Risk and Return II

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Q8.

Which of the following risks is priced in capital market theory?

A. Market risk

B. Nonsystematic risk

C. Total risk

查看答案与解析

答案：A

解析：在资本市场理论中，只有市场风险（系统性风险）被定价。非系统性风险可以通过分散化消除，因此市场不会为其提供补偿。总风险包括系统性和非系统性风险，但只有系统性部分被定价。

选项	判断	解析
A	✓	市场风险（系统性风险）是唯一被定价的风险
B	✗	非系统性风险可以分散化消除，不被定价
C	✗	总风险中只有系统性部分被定价

关联：R84: Portfolio Risk and Return II

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Q9.

As the number of stocks in the portfolio increases, the unsystematic risk of a portfolio tends to:

A. increase at a decreasing rate.

B. decrease at a decreasing rate.

C. decrease at an increasing rate.

查看答案与解析

答案：B

解析：随着组合中股票数量增加，非系统性风险逐渐减少，但减少的速度越来越慢（以递减的速率下降）。最初增加几只股票时风险显著降低，之后边际分散化效果递减。系统性风险始终存在，无法通过增加股票数量消除。

选项	判断	解析
A	✗	非系统性风险是递减的，不是递增的
B	✓	非系统性风险以递减的速率下降（边际分散化效果递减）
C	✗	下降速率是递减的，不是递增的

关联：R84: Portfolio Risk and Return II

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Q10.

Regarding the return generating models, the slope term and the intercept term of the market model are:

	Slope term	Intercept term
A.	Systematic risk	Alpha
B.	Unsystematic risk	Beta
C.	Total risk	Alpha

A. A

B. B

C. C

查看答案与解析

答案：A

解析：市场模型为 $R_i={\alpha }_i+{\beta }_i{R}_m+e_i$。其中截距项是 alpha（$\alpha$），斜率项是 beta（$\beta$），beta 衡量的是系统性风险。选项A正确识别了斜率项代表系统性风险，截距项代表 alpha。

选项	判断	解析
A	✓	斜率（beta）= 系统性风险衡量，截距 = alpha
B	✗	斜率不是非系统性风险，截距不是 beta
C	✗	斜率不是总风险

关联：R84: Portfolio Risk and Return II

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Q11.

A return-generating model that provides an estimate of the expected return of a security based on factors such as earnings growth and cash flow generation is best described as a:

A. market factor model.

B. fundamental factor model.

C. macroeconomic factor model.

查看答案与解析

答案：B

解析：基本面因子模型（fundamental factor model）使用公司层面的基本面因素（如盈利增长、现金流生成、市盈率等）来解释和预测证券回报。市场因子模型只使用市场组合回报作为因子。宏观经济因子模型使用GDP增长、通胀等宏观经济变量。

选项	判断	解析
A	✗	市场因子模型仅使用市场回报作为单一因子
B	✓	基本面因子模型使用公司层面的因素如盈利增长和现金流
C	✗	宏观经济因子模型使用宏观经济变量如GDP、通胀等

关联：R84: Portfolio Risk and Return II

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Q12.

Which of the following statements about return-generating models is/are correct?

Statement 1: The intercept of the market model is the asset’s estimated beta.

Statement 2: Return-generating models are used to estimate the expected return of a security.

A. Statement 1.

B. Statement 2.

C. Both of them.

查看答案与解析

答案：B

解析：市场模型 $R_i={\alpha }_i+{\beta }_i{R}_m+e_i$ 的截距项是 alpha（$\alpha$），而非 beta。Beta 是斜率项。Statement 1 错误。Statement 2 正确：回报生成模型的核心用途就是估计证券的预期回报。

选项	判断	解析
A	✗	Statement 1 错误：截距是 alpha，不是 beta
B	✓	Statement 2 正确：回报生成模型用于估计预期回报
C	✗	Statement 1 是错误的

关联：R84: Portfolio Risk and Return II

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Q13.

A newly issued stock with a standard deviation of 30% has a correlation of 0.6 with the market. Assuming that the standard deviation of the market is 15%, the beta of the stock is closest to:

A. 1.2

B. 0.3

C. 0.027

查看答案与解析

答案：A

解析：Beta 的计算公式为：

计算过程：

${\beta }_i=\frac{{\rho }_{i,m}\times {\sigma }_i}{{\sigma }_m}=\frac{0.6\times 0.30}{0.15}=\frac{0.18}{0.15}=1.2$

选项	判断	解析
A	✓	$\beta =0.6\times 0.30/0.15=1.2$
B	✗	可能是 \times 0.30 / 0.60$ 的错误计算
C	✗	可能是使用了方差而非标准差的错误计算

关联：R84: Portfolio Risk and Return II

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Q14.

The expected market return is 6% and the risk-free rate is 2%. With respect to the capital asset pricing model, if the expected return for Security A is 10%, the beta of Security A is closest to:

A. 1.

B. 2.

C. 3.

查看答案与解析

答案：B

解析：使用 CAPM 公式反推 beta：

计算过程：

$E(R_A)=R_f+{\beta }_A[E(R_m)-R_f]$

$10% = 2% + \beta_A (6% - 2%)$$

$8% = \beta_A \times 4%$$

${\beta }_A=\frac{8\%}{4\%}=2$

选项	判断	解析
A	✗	如果 $\beta =1$，则 $E(R)=2\%+4\%=6\%$，不是 10%
B	✓	$\beta =2$：$E(R)=2\%+2\times 4\%=10\%$
C	✗	如果 $\beta =3$，则 $E(R)=2\%+12\%=14\%$，不是 10%

关联：R84: Portfolio Risk and Return II

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Q15.

There are two risky assets held by ABC Fund, and the detailed information is shown in the table:

	Market Value (in million)	Beta
Asset I	3.5	1.4
Asset II	7.5	1.8

The total market value of the fund is currently 12 million, and the rest of the investable capital is deposited in a bank account to earn a risk-free return of 1.5%. The fund’s beta is closest to:

A. 1.53

B. 1.67

C. 1.66

查看答案与解析

答案：A

解析：组合 beta 是各资产 beta 的加权平均。银行存款（无风险资产）的 beta = 0。

计算过程：

${\beta }_p=\frac{3.5}{12}\times 1.4+\frac{7.5}{12}\times 1.8+\frac{1.0}{12}\times 0$

$=0.2917\times 1.4+0.625\times 1.8+0$

$=0.4083+1.125=1.5333\approx 1.53$

注：剩余资本 = 12 - 3.5 - 7.5 = 1.0 百万

选项	判断	解析
A	✓	正确计算包含无风险资产（beta = 0）的加权平均 beta
B	✗	可能只计算了两个风险资产的加权 beta，未考虑现金
C	✗	计算错误

关联：R84: Portfolio Risk and Return II

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Q16.

Which of the following statements is a correct assumption of capital asset pricing model (CAPM)?

A. All investors are price takers.

B. All investors have multi-period investment horizons.

C. All investors can purchase the assets with a certain level of unit.

查看答案与解析

答案：A

解析：CAPM 的假设包括：(1) 所有投资者都是价格接受者（price takers）；(2) 所有投资者具有相同的单期投资期限（single-period，而非 multi-period）；(3) 投资是无限可分的（infinitely divisible），即可以购买任意比例的资产。

选项	判断	解析
A	✓	所有投资者是价格接受者是 CAPM 的假设
B	✗	CAPM 假设单期投资期限，而非多期
C	✗	CAPM 假设资产无限可分，不需要特定单位

关联：R84: Portfolio Risk and Return II

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Q17.

When we use the capital asset pricing model (CAPM), the most accurate assumption is that:

A. all investments are infinitely divisible and always sufficient in the market.

B. investors have the homogeneous expectation of future market trends even if some of them will be irrational in extreme market situations.

C. Costs and taxes have impacts and adjustments should be taken into consideration in the model.

查看答案与解析

答案：A

解析：CAPM 假设所有投资无限可分且市场中始终有充足的供应。CAPM 假设所有投资者都是理性的（不存在非理性行为），且不考虑交易成本和税收。

选项	判断	解析
A	✓	所有投资无限可分且市场供应充足是 CAPM 的假设
B	✗	CAPM 假设所有投资者都是理性的，不允许非理性行为
C	✗	CAPM 假设无交易成本和无税收

关联：R84: Portfolio Risk and Return II

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Q18.

An asset’s return has a standard deviation of 0.1 and a beta of -0.5. If the market risk premium is 7% and the risk-free rate is 4%, based on the capital asset pricing model, the expected return of the asset is closest to:

A. 0.5%.

B. 4.7%.

C. 2.5%.

查看答案与解析

答案：A

解析：使用 CAPM 公式计算预期回报。标准差在此题中是干扰信息。

计算过程：

$E(R)=R_f+\beta \times [E(R_m)-R_f]=4\%+(-0.5)\times 7\%=4\%-3.5\%=0.5\%$

注意：市场风险溢价（market risk premium）= $E(R_m)-R_f=7\%$

选项	判断	解析
A	✓	\% + (-0.5) \times 7% = 0.5%$
B	✗	可能忽略了 beta 的负号
C	✗	计算错误

关联：R84: Portfolio Risk and Return II

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Q19.

The slope of the security market line (SML) represents the concept of:

A. excess market return over the risk-free rate.

B. return premium compensated for undertaking one unit of total risk.

C. return premium compensated for undertaking one unit of systematic risk.

查看答案与解析

答案：C

解析：SML 的公式为 $E(R_i)=R_f+{\beta }_i[E(R_m)-R_f]$。SML 的斜率是市场风险溢价 $[E(R_m)-R_f]$，代表每承担一单位系统性风险（beta）所获得的额外回报补偿。注意区分：CML 的斜率是每单位总风险的补偿（夏普比率），而 SML 的斜率是每单位系统性风险的补偿。

选项	判断	解析
A	✗	市场超额回报是斜率的数值，但这不完整地描述其含义
B	✗	SML 衡量的是系统性风险，不是总风险（总风险对应 CML）
C	✓	SML 斜率 = 每单位系统性风险（beta）的回报补偿

关联：R84: Portfolio Risk and Return II

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Q20.

Which of the following statements about capital market line and security market line is/are correct?

Statement 1: Capital market line is used for security selection.

Statement 2: The slope of the security market line is market risk premium.

A. Statement 1.

B. Statement 2.

C. Both of them.

查看答案与解析

答案：B

解析：CML 用于资产配置（asset allocation），而 SML 用于证券选择（security selection）。SML 的斜率确实是市场风险溢价 $[E(R_m)-R_f]$。

选项	判断	解析
A	✗	CML 用于资产配置，SML 才用于证券选择
B	✓	SML 的斜率 = 市场风险溢价 = $E(R_m)-R_f$
C	✗	Statement 1 错误

关联：R84: Portfolio Risk and Return II

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Q21.

BuildingBlue, a real estate company is evaluating the economic feasibility of a new 2-year project. The initial investment in Year 1 is $250 million. The incomes from the project at the end of Year 1 and Year 2 are $100 million and $200 million, respectively. The market return is 8% and the risk-free rate is 3%. Assuming the beta of the project is 2.5, the project:

A. should be accepted by BuildingBlue, as the net present value is positive.

B. should be rejected by BuildingBlue, as the net present value is positive.

C. should not be accepted by BuildingBlue, as the net present value is negative.

查看答案与解析

答案：C

解析：首先用 CAPM 计算项目的要求回报率，再计算 NPV。

计算过程：

$r=R_f+\beta (R_m-R_f)=3\%+2.5\times (8\%-3\%)=3\%+12.5\%=15.5\%$

$NPV=-250+\frac{100}{1.155}+\frac{200}{(1.155{)}^2}$

$=-250+86.58+149.93=-13.49\text{ million}$

NPV 为负，项目不应被接受。

选项	判断	解析
A	✗	NPV 为负，不是正的
B	✗	说法自相矛盾（NPV正但应拒绝）
C	✓	NPV 约为 -13.49 百万，为负值，项目应被拒绝

关联：R84: Portfolio Risk and Return II

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Q22.

If the estimated return of an asset lies below the security market line, the asset should be:

A. overpriced.

B. properly priced.

C. underpriced.

查看答案与解析

答案：A

解析：SML 上的点代表合理定价的资产。如果资产的预期回报低于 SML 给出的回报（即位于 SML 下方），说明该资产的实际预期回报不足以补偿其系统性风险，因此该资产被高估（overpriced），投资者应卖出。

选项	判断	解析
A	✓	低于 SML = 预期回报不足 = 价格过高 = overpriced
B	✗	只有位于 SML 上的资产才是合理定价
C	✗	低估的资产应位于 SML 上方（预期回报高于应有水平）

关联：R84: Portfolio Risk and Return II

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Q23.

The stock has a beta of 1.6 and is currently priced at $30. It is anticipated to rise to $32 by the end of the year and also provide a dividend of $0.5. Assuming an expected market return of 8% and a risk-free rate of 5%, the stock is

A. overpriced.

B. properly priced.

C. underpriced.

查看答案与解析

答案：A

解析：比较实际预期回报和 CAPM 要求回报率。

计算过程：

实际预期回报： $R_{actual}=\frac{(32-30)+0.5}{30}=\frac{2.5}{30}=8.33\%$

CAPM 要求回报率： $R_{required}=5\%+1.6\times (8\%-5\%)=5\%+4.8\%=9.8\%$

实际回报（8.33%）< 要求回报（9.8%），说明股票被高估。

选项	判断	解析
A	✓	实际预期回报 8.33% < CAPM 要求回报 9.8%，股票被高估
B	✗	两个回报率不相等，不是合理定价
C	✗	低估需要实际回报 > 要求回报

关联：R84: Portfolio Risk and Return II

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Q24.

Three equity fund managers have their performance records summarized in the table below.

Fund Manager	Return	Standard Deviation
Manager 1	10%	15%
Manager 2	12%	18%
Manager 3	8%	12%

Considering a risk-free rate of return of 2.50%, which manager demonstrated the highest performance according to the Sharpe ratio?

A. Manager 1.

B. Manager 2.

C. Manager 3.

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答案：B

解析：夏普比率 = (回报 - 无风险利率) / 标准差。

计算过程：

$Sharp{e}_1=\frac{10\%-2.5\%}{15\%}=\frac{7.5\%}{15\%}=0.500$

$Sharp{e}_2=\frac{12\%-2.5\%}{18\%}=\frac{9.5\%}{18\%}=0.528$

$Sharp{e}_3=\frac{8\%-2.5\%}{12\%}=\frac{5.5\%}{12\%}=0.458$

Manager 2 的夏普比率最高。

选项	判断	解析
A	✗	Manager 1 夏普比率 = 0.500
B	✓	Manager 2 夏普比率 = 0.528，最高
C	✗	Manager 3 夏普比率 = 0.458

关联：R84: Portfolio Risk and Return II

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Q25.

Which of the following performance measures uses total risk rather than systematic risk?

A. Sharpe ratio

B. Treynor ratio

C. Jensen’s alpha

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答案：A

解析：夏普比率使用标准差（总风险）作为风险衡量指标。特雷诺比率使用 beta（系统性风险）。詹森 alpha 也基于 beta（系统性风险）来计算。

选项	判断	解析
A	✓	夏普比率 = $(R_p-R_f)/{\sigma }_p$，分母是标准差（总风险）
B	✗	特雷诺比率 = $(R_p-R_f)/{\beta }_p$，分母是 beta（系统性风险）
C	✗	詹森 alpha = $R_p-[R_f+{\beta }_p(R_m-R_f)]$，基于 beta

关联：R84: Portfolio Risk and Return II

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Q26.

Which of the following performance measurements is most suitable for a well-diversified portfolio?

A. Treynor ratio

B. Sharpe ratio

C. M-squared

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答案：A

解析：对于充分分散化的组合，非系统性风险已被消除，因此组合的总风险近似等于系统性风险。在这种情况下，使用基于系统性风险（beta）的特雷诺比率更为合适。夏普比率和 M-squared 使用总风险，对于充分分散化的组合来说，两种方法的排名结果可能相似，但特雷诺比率更直接衡量系统性风险的补偿效率。

选项	判断	解析
A	✓	充分分散化组合中非系统性风险几乎为零，使用基于 beta 的特雷诺比率最合适
B	✗	夏普比率使用总风险，对充分分散化组合也适用但非最佳选择
C	✗	M-squared 基于总风险，与夏普比率类似

关联：R84: Portfolio Risk and Return II